Production Function: Law of Variable Proportions and Law of Returns to Scale:
Introduction:
In traditional production theory resources used for the production of a product are known as factors of production. Factors of production are now termed as inputs which may mean the use of the services of land, labour, capital and organization in the process of production. The term output refers to the commodity produced by the various inputs.
Production theory concerns itself with the problems of combining various inputs, given the state of technology, in order to produce a stipulated output. The technological relationships between inputs and outputs are known as production functions.
Production:
Production in economic, terms is generally understood as the transformation of inputs into outputs. The inputs are what the firm buys, namely productive resources, and outputs are what it sells. Production is not the creation of matter but it is the creation of value. Production is also defined as producing goods which satisfy some human want. Production is a sequence of technical processes requiring either directly or indirectly the mental and physical skill of craftsman and consists of changing the shape, size and properties of materials and ultimately converting them into more useful articles.
Methods of Production:
There are three methods of production:
a) Unit production
b) Mass production
c) Batch production
The unit production is otherwise known as job-order production. This type of production is used for things which cannot be produced on large scale, things of high artistic nature, i.e. production of exclusive goods. This is a method to meet the individual requirements of customers. This type of production requires lot of flexibility in operation.
Mass production uses mechanical aids for material handling. This type of production requires specially planned layout, special purpose machines, jigs and fixtures, automatic machines, etc. Mass production is continuous production, i.e. it does not have any non-producing time.
Batch production is generally adopted in medium size enterprises. It is a stage in-between unit production and mass production. It is bigger in scale than unit production while it is smaller than mass production. In this type of production, variety of products is manufactured in lots at regular interval. Therefore, this is known as batch production. The theory of production centres round the concept of production function which we explain now.
The Production Function:
The production function expresses a functional relationship between quantities of inputs and outputs. It shows how and to what extent output changes with variations in inputs during a specified period of time. In the words of Stigler, “The production function is the name given to the relationship between rates of input of productive services and the rate of output of product.
It is the economist’s summary of technical knowledge.” Basically, the production function is a technological or engineering concept which can be expressed in the form of a table, graph and equation showing the amount of output obtained from various combinations of inputs used in production, given the state of technology. Algebraically, it may be expressed in the form of an equation as
Q =f (L, M, N, К, T)…………. (1)
where Q stands for the output of a good per unit of time, L for labour, M for management (or organisation), N for land (or natural resources), К for capital and T for given technology, and refers to the functional relationship.
The production function with many inputs cannot be depicted on a diagram. Moreover, given the specific values of the various inputs, it becomes difficult to solve such a production function mathematically. Economists, therefore, use a two-input production function. If we take two inputs, labour and capital, the production function assumes the form
Q = f (L, K) ….(2)
The production function as determined by technical conditions of production is of two types:
It may be rigid ox flexible. The former relates to the short run and the latter to the long run.
The Nature of Production Function:
The production function depends upon the following factors:
(a) The quantities of inputs to be used.
(b) The state of technical knowledge.
(c) The possible processes of production.
(d) The size of the firm.
(e) The prices of inputs.
Now if these factors change the production function automatically changes.
Attributes of Production Function:
The following are the important attributes of production function:
(i) The production function is a flow concept.
(ii) A production function is a technical relationship between inputs and outputs expressed in physical terms.
(iii) The production function of a firm depends on the state of technology and inputs.
(iv) From the economic point of view, a rational firm is interested not in all the numerous possible levels of output but only in that combination which yields maximum outputs.
(v) The short run production function pertains to the given scale of production. The long run production function pertains to the changing scale of production.
The Short-Run Production Function:
In the short run, the technical conditions of production are rigid so that the various inputs used to produce a given output are in fixed proportions. However, in the short run, it is possible to increase the quantities of one input while keeping the quantities of other inputs constant in order to have more output. This aspect of the production function is known as the Law of Variable Proportions. The short-run production function in the case of two inputs, labour and capital, with capital as fixed and labour as the variable input can be expressed as
Q=f (L,K)
where K refers to the fixed input. … (3)
This production function is depicted in Figure 1 where the slope of the curve shows the marginal product of labour. A movement along the production function shows the increase in output as labour increases, given the amount of capital employed K;. If the amount of capital increases to K, at a point of time, the production function Q = f (L, K 1) shifts upwards to Q = f (L,K2 ), as shown in the figure.
On the other hand, if labour is taken as a fixed input and capital as the variable input, the production function takes the form Q =f (KL) …(4)
This production function is depicted in Figure 2 where the slope of the curve represents the marginal product of capital. A movement along the production function shows the increase in output as capital increases, given the quantity of labour employed, L2 If the quantity of labour increases to L2 at a point of time, the production function Q = f (K,L 1) shifts upwards to Q=f(KL2).
The Long-Run Production Function:
In the long run, all inputs are variable. Production can be increased by changing one or more of the inputs. The firm can change its plants or scale of production. Equations (1) and (2) represent the long-run production function. Given the level of technology, a combination of the quantities of labour and capital produces a specified level of output.
The long-run production function is depicted in Figure 3 where the combination of OK of capital and OL of labour produces 100 Q. With the increase in inputs of capital and labour to OK1 and OL1, the output increases to 200 Q. The long-run production function is shown in terms of an isoquant such as 100 Q.
In the long run, it is possible for a firm to change all inputs up or down in accordance with its scale. This is known as returns to scale. The returns to scale are constant when output increases in the same proportion as the increase in the quantities of inputs. The returns to scale are increasing when the increase in output is more than proportional to the increase in inputs. They are decreasing if the increase in output is less than proportional to the increase in inputs.
Let us illustrate the case of constant returns to scale with the help of our production function.
Q = (L, M, N, К, T)
Given T, if the quantities of all inputs L, M, N, K are increased n-fold, the output Q also increases и-fold. Then the production function becomes nQ –f (nL, nM, nN, nK).
This is known as linear and homogeneous production function, or a homogeneous function of the first degree. If the homogeneous function is of the Kth degree, the production function is nk.Q = f (nL, nM, nN, nK) If k is equal to 1, it is a case of constant returns to scale; if it is greater than 1, it is a case of increasing returns of scale; and if it is less than 1, it is a case of decreasing returns to scale.
Thus a production function is of two types:
(i) Linear homogeneous of the first degree in which the output would change in exactly the same proportion as the change in inputs. Doubling the inputs would exactly double the output, and vice versa. Such a production function expresses constant returns to scale,
(ii) Non-homogeneous production function of a degree greater or less than one. The former relates to increasing returns to scale and the latter to decreasing returns to scale.
Conclusion:
The production function exhibits technological relationships between physical inputs and outputs and is thus said to belong to the domain of engineering. Prof. Stigler does not agree with this commonly held view. The function of management is to sort out the right type of combination of inputs for the quantity of output he desires.
For this, he has to know the prices of his inputs and the technique to be used for producing a specified output within a specified period of time. All these technical possibilities
are derived from applied sciences, but cannot be worked out by technologists or engineers alone. ‘The entrepreneurs also provide productive services and they are far from standardized.
are derived from applied sciences, but cannot be worked out by technologists or engineers alone. ‘The entrepreneurs also provide productive services and they are far from standardized.
Some men can get gang of workers to do their best, others are better at luring customers, still others at borrowing money, and each will have a different production function. If we take account of activities such as selling, settling strikes and anticipating future styles of product, it is clear that large segments of what we mean by technique are matters of business knowledge and talents, not to be acquired in the best engineering schools.” The production function is, in fact, “the economist’s summary of technological knowledge,” as pointed out by Prof. Stigler.
The Law of Variable Proportions:
If one input is variable and all other inputs are fixed the firm’s production function exhibits the law of variable proportions. If the number of units of a variable factor is increased, keeping other factors constant, how output changes is the concern of this law. Suppose land, plant and equipment are the fixed factors, and labour the variable factor.
When the number of labourers is increased successively to have larger output, the proportion between fixed and variable factors is altered and the law of variable proportions sets in. The law states that as the quantity of a variable input is increased by equal doses keeping the quantities of other inputs constant, total product will increase, but after a point at a diminishing rate.
This principle can also be defined thus:
When more and more units of the variable factor are used, holding the quantities of fixed factors constant, a point is reached beyond which the marginal product, then the average and finally the total product will diminish. The law of variable proportions (or the law of non-proportional returns) is also known as the law of diminishing returns. But, as we shall see below, the law of diminishing returns is only one phase of the more comprehensive law of variable proportions.
Its Assumption:
The law of diminishing returns is based on the following assumptions:
(1) Only one factor is variable while others are held constant.
(2) All units of the variable factor are homogeneous.
(3) There is no change in technology.
(4) It is possible to vary the proportions in which different inputs are combined.
(5) It assumes a short-run situation, for in the long-run all factors are variable.
(6) The product is measured in physical units, i.e., in quintals, tonnes, etc. The use of money in measuring the product may show increasing rather than decreasing returns if the price of the product rises, even though the output might have declined.
Its Explanation:
Given these assumptions, let us illustrate the law with the help of Table 1, where on the fixed input land of 4 acres, units of the variable input labour are employed and the resultant output is obtained. The production function is revealed in the first two columns. The average product and marginal product columns are derived from the total product column.
The average product per worker is obtained by dividing column (2) by a corresponding unit in column (1). The marginal product is the addition to total product by employing an extra worker. 3 workers produce 36 units and 4 produce 48 units. Thus the marginal product is 12 i.e., (48-36) units.
An analysis of the Table shows that the total, average and marginal products increase at first, reach a maximum and then start declining. The total product reaches its maximum when 7 units of labour are used and then it declines. The average product continues to rise till the 4th unit while the marginal product reaches its maximum at the 3rd unit of labour, then they also fall. It should be noted that the point of falling output is not the same for total, average and marginal product.
The marginal product starts declining first, the average product following it and the total product is the last to fall. This observation points out that the tendency to diminishing returns is ultimately found in the three productivity concepts.
The law of variable proportions is presented diagrammatically in Figure 4. The TP curve first rises at an increasing rate up to point A where its slope is the highest. From point A upwards, the total product increases at a diminishing rate till it reaches its highest point С and then it starts falling.
Point A where the tangent touches the TP curve is called the inflection point up to which the total product increases at an increasing rate and from where it starts increasing at a diminishing rate. The marginal product curve (MP) and the average product curve (AP) also rise with.
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